You should also have no difficulty using Pythagoras to check that A = (2,√(11)), B = (0,√3), C = (2,√3).
For (iii), the conformal property of the hyperbolic space tells you that the hyperbolic angles are the same as the Cartesian angles. You can get those by calculus from the Cartesian equations of the circles, finding the slope of the tangents to the circles at B and C, and using some trigonometry.
For (iv), use the formula for the hyperbolic length given here (together with the formula giving the inverse cosh in terms of a natural logarithm).
Edit. In fact, you don't need calculus for (iii). Just use the fact that the tangent is perpendicular to the radius. You can read off the slope of the radius directly from the diagram.